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There are two types of dirac equations: $(p_\mu\gamma^\mu - m)\Psi(x) = 0$ and $(p_\mu\gamma^\mu + m)\Psi(x) = 0$. Here $p$ are the momentum operators.

The fermion propagator is defined in the equation $(p_\mu\gamma^\mu - m)S_F(x-x') = \delta(x-x')$. This equation can be easily solved in momentum space: $$S_F(p) = \frac{p_\mu\gamma^\mu + m}{p^2-m^2+i\epsilon}$$.

As far as I can see this contains only the first dirac equation and not the second equation. Nevertheless we use this to calculate all processes in QED, so is the seconed dirac equation not included in QED?

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The two equations would lead to equivalent physics but only one of them may be right: the correct equations of motion for Dirac spinor fields are first-order in derivatives. The convention is that only the first equation is right for the Dirac field $\Psi$. The second equation is simply incorrect and doesn't follow from the Lagrangian etc.

If we wanted solutions to both equations to be right, $\Psi$ would effectively obey the Klein-Gordon equation of the type $(p^2-m^2) \Psi = 0$ for each component. But a virtue of the Dirac equation is that it is a first-order equation. That's also why it can be so nicely reduced to the non-relativistic Schrödinger's equation in the non-relativistic limi5 which is also first-order in time derivatives. This reduction is needed for the Dirac equation to make the right predictions for the hydrogen atom, among related things.

If $\Psi$ obeys the first equation, then e.g. $(-i) (\bar \Psi \gamma^0 \gamma^2)^T$, the charge-conjugate Dirac spinor, obeys the second equation. So it's easy to produce solutions of the second equation from the solution of the first equation (the two charge-conjugate fields are related in the same way as particles and antiparticles) but only one sign of the mass of a Dirac field is right once we fix the conventions!

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Mathematically speaking, the first equation for the Dirac spinor field $Ψ$ is the same as the second equation for the Dirac spinor field $γ_5 Ψ$ where $γ_5$ is the parity-odd gamma matrix, so yes, they are both correct, both follow from the same Lagrangian, both have the same conserved quantities (energy and spin, as well as all current densities); in the non-relativistic limit, you would still obtain that two components of the spinor would vanish, the only difference being that for the first form the two components to vanish are the two lower ones while for the second form the two components to vanish are the two upper ones (in standard representation), but you would still get the same Schrödinger equation for the remaining components: so, both signs of the mass are right and both forms are correct, because they are symmetric under the transformation $m →-m$ and $Ψ → γ_5 Ψ$ always possible.

But this is the problem, because since such a transformation is always possible but it does not change anything at all, then you may say the new fields give rise to nothing new, and therefore it is enough to consider only one of them, any one of them would do, and the choice fell on the first.

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